Field of the Invention
This invention relates to a database for polytope and convex bodies, possibly with integral constraints.
Discussion of Prior Art
Every convex optimization problem has an objective function and one or more convex constraints. Efficient representation of convex constraints and sets of constraints is central for high performance. Non-degenerate convex constraint sets results in N-dimensional regions of non-zero volume, which offers new possibilities for database design and optimization.
Early work on constraint databases deals with GIS query processing, but does not discuss mathematical programming aspects. Other previous work, in this field describes constraint attribute systems, but none in a mathematical programming context. The rich structure of mathematical programming problems enables many unique features in this database, as opposed to general constraint programming systems.
Most of the previous work refers to polynomial time solvability of query expressions in First-order and higher logics, over polynomial constraints, but applications to uncertainty, or experimental runtimes were not reported. In addition, the rich mathematical theory of convex optimization including duality is not referred to.
U.S. Ser. No. 13/255,408 titled “New vistas in inventory optimization under uncertainty” describes a computer implemented method for carrying out inventory optimization under uncertainty. The method involves the step of feeding the information in the form of polyhedral formulation of uncertainty where the faces and edges of polytope are built from linear constraints that are derived from historical time series data. This approach leads to a generalization of basestock policies to multidimensional correlated variables which can be used in many contexts [1].
U.S. Ser. No. 13/003,507 titled “A computer implemented decision support method and system” describe a decision support method which extends on the robust optimization technique. The method involves the representation of the uncertainty as polyhedral uncertainty sets made of simple linear constraints derived from macroscopic economic data. The constraint sets are has pre-set and allowable parameters. It is applied in the field of capacity planning and inventory optimization problems in supply chains [1].
WO 2010/004585 titled “Decision support methods under uncertainty” describes a computer implemented method of handling uncertainty or hypothesis about operating systems by use of probabilistic formulation and constraints based method. The method finds the set theoretic relationship-subset, intersection and disjointness among the polytopes and proposes a method to visualize the relationship. This helps in the decision support for the relationship between the said constraint sets of the polytopes.